Let A be a dense of [a,b] and let f:[a,b]-->R be an integrable function such that f(x)>=0 for every element of x in A. Prove that the (integral of a to b) f >= 0.

Please help. I have ideas for this but need help solving

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- April 1st 2010, 08:06 AMderek walcottDense subset
Let A be a dense of [a,b] and let f:[a,b]-->R be an integrable function such that f(x)>=0 for every element of x in A. Prove that the (integral of a to b) f >= 0.

Please help. I have ideas for this but need help solving - April 1st 2010, 09:26 AMTinyboss
I assume you mean f is Riemann-integrable--it isn't true for Lebesgue integration.

For any partition P, every interval will contain an element of A, and therefore the upper sum is non-negative for that partition. Since f is integrable, the upper sums converge to the integral as the mesh goes to zero, and so the integral is non-negative. - April 1st 2010, 09:47 AMDrexel28